Bending of BeamsMECHENG242 Mechanics of Materials

www.engineering.auckland.ac.nz/mechanical/MechEng242

2.0 Bending of BeamsNow we consider the elastic deformation of beams (bars) under bending loads.

1.0 Axial Forces☻

MM

Bending of BeamsMECHENG242 Mechanics of Materials

Application to a Bar

Normal Force:

FnFn

Shear Force:

FtFt

Bending Moment:

MtMt

Torque or Twisting Moment:

Mn

Mn

S.B.

K.J.

Bending of BeamsMECHENG242 Mechanics of Materials

Examples of Devices under Bending Loading:

Car Chassis

YachtExcavator

Atrium Structure

Bending of BeamsMECHENG242 Mechanics of Materials

2.2 Stresses in Beams (Refer: B,C & A –Sec’s 6.3-6.6)

2.3 Combined Bending and Axial Loading (Refer: B,C & A –Sec’s 6.11, 6.12)

2.1 Revision – Bending Moments (Refer: B,C & A – Sec’s 6.1,6.2)

2.0 Bending of Beams

2.4 Deflections in Beams (Refer: B,C & A –Sec’s 7.1-7.4)

2.5 Buckling (Refer: B,C & A –Sec’s 10.1, 10.2)

x

x

Mxz Mxz

x

P

P1

P2

Bending of BeamsMECHENG242 Mechanics of Materials

2.1 Revision – Bending Moments

Last year Jason Ingham introduced Shear Force and Bending Moment Diagrams.

(Refer: B, C & A – Chapter 6)

3m 3m

12 kN

RECALL…

Q

0

(SFD)

0

M (BMD)

Bending of BeamsMECHENG242 Mechanics of Materials

(Refer: B, C&A – Sections 1.14, 1.15, 1.16, 6.1)

x

y

PA B

RByRAyMxz Mxz

Radius of Curvature, R

Deflected Shape

Consider the simply supported beam below:

Mxz MxzWhat stresses are generated within, due to bending?

Bending of BeamsMECHENG242 Mechanics of Materials

Axial Stiffness

Load (W)

Extension (u)

Bending Moment

(Mxz)

Curvature (1/R)

Flexural Stiffness

PA B

RByMxzRAy

Mxz

BendingRecall: Axial Deformation

W

u

Bending of BeamsMECHENG242 Mechanics of Materials

x

y

Mxz=Bending Moment

Mxz Mxz

Beam

x (Tension)

x (Compression)

x=0

(i) Bending Moment, Mxz (ii) Geometry of Cross-section

x is NOT UNIFORM through the section depth

x DEPENDS ON:

Axial Stress Due to Bending:

Unlike stress generated by axial loads, due to bending:

Bending of BeamsMECHENG242 Mechanics of Materials

“Happy” Beam is +VE “Sad” Beam is -VE

x

yMxz=Bending Moment

+VE (POSITIVE)

Sign Conventions: Qxy=Shear Force

Mxz Mxz

Qxy Qxy

-ve x

+ve x

Bending of BeamsMECHENG242 Mechanics of Materials

;0Fy ;0Mz

PQxy

x

yExample 1: Bending Moment Diagrams P

RAy=P

A B

L

Mxz=P.L

P.L

P Qxy

Mxz

xP

Mxz

QxyMxz Mxz

Qxy Qxy

Q & M are POSITIVE xLPMxz

Bending of BeamsMECHENG242 Mechanics of Materials

;PQxy

x

yP

P

BP.L

x

xLPMxz L

Qxy 0

Mxz 0

A Mxz

Qxy

To find x and deflections, need to know Mxz.

Shear Force Diagram (SFD)

Bending Moment Diagram (BMD)

+veP

-ve

-P.L

Bending of BeamsMECHENG242 Mechanics of Materials

x

yExample 2: Macaulay’s Notation

Qxy

Mxz

A BC

a bP

baaPRBy

ba

bPRAy

x

babP

;0Mz xzM

axPxba

PbMxz

Where can only be +VE or ZERO. ax

Pa

A

axP 0xba

Pb

Bending of BeamsMECHENG242 Mechanics of Materials

x

y

x

A BC

a bP

baPa ba

Pb

(i) When

:ax

(ii) When

:ax axPxba

PbMxz

1

2

axPxba

PbMxz

0

A BC

+ve baPab

Mxz

0

BMD: Eq. 1Eq. 2

Bending of BeamsMECHENG242 Mechanics of Materials

;0Fy ;0Mz

x

y

Example 3: Distributed Load

RAy=wL

A B

Lx

Qxy

Mxz

Mxz

Qxy

Mxz=wL2

2

wL

wL2

2

Distributed Load w per unit length

wL

2wLM

2

xz

xLwQxy

wx

wx 0Qxy

xwL 02xwx

Bending of BeamsMECHENG242 Mechanics of Materials

;0x@

2wL

2wxwLxM

22

xz xLwQxy

-ve

-wL2

2

x

Mxz

0BMD: L

2wLM

2

xz

;Lx@ 0Mxz

;2Lx@ 8

wLM2

xz

Bending of BeamsMECHENG242 Mechanics of Materials

Summary – Is anything Necessary for RevisionGenerating Bending Moment Diagrams is a key skill you must revise. From these we will determine:

• Stress Distributions within beams,

• and the resulting Deflections

Apart from the revision problems on Sheet 4, you might try these sources:

• B, C & A Worked Examples, pg 126-132 Problems, 6.1 to 6.8, pg 173

• Jason Ingham’s problem sheets: www.engineering.auckland.ac.nz/mechanical/EngGen121